Motivation

The flow around an infinite, circular cylinder is a well-studied, but yet not completely understood problem in CFD (computational fluid dynamics). A simple exploration of the steady and unsteady flow at low Reynolds numbers is undertaken here, to provide a light overview of the problem.

Mesh Assembly : Preliminaries

We assemble a set of preliminary meshes for simulations with Reynolds number 20 and 110, consisting of an inner cylindrical segment and outer rectangular segment as follows. The following mesh schematics list dimension parameters and block configuration, courtesy of Dr.Fabrizio Bisetti, UT Austin.

Preliminary Solutions

We assemble a set of preliminary solutions for Reynolds number 20 and 110 (the previous meshes). The Re=20 mesh will be shown first.
Three contour plots will be shown: \(\frac{u}{U}\), \(\frac{v}{U}\), \(\frac{p}{\rho U^2}\), along with a streamline plot.
For Re=110 (the unsteady case), a time history of \(\frac{u}{U}\), \(\frac{v}{U}\), \(\frac{p}{\rho U^2}\) at \((x,y)=(5.5,\pm0.5)\) versus normalized time \(\frac{t}{\frac{D}{U}}\).

Note data for a couple of points (about 0.1%) was imputed due to file errors.

• run_20_1:

• run_110_1:

Time History plots for run_110_1:

Mesh Improvement

We can see that the mesh extent is fairly adequate, except possibly the height (H) for Re=110. All boundary flow is not perturbed, with the exception of the aforementioned region. We can also see vortices for the Re=110 case, and the inner block does not cause any boundary issues, so we will make only the following minor changes to each mesh.

For the mesh run_20_1, we will increase the blockMeshFactor from 10 to 15 (so a block increase of ~2.25). This mesh will be called run_20_2.

For the mesh run_110_1, we will increase height (H) from 4 to 6. This mesh will be called run_110_2.

Unsteady Flow and Strouhal Number

We now use the simulations run_110_3, run_110_4, and run_110_2 to calculate an approximate Strouhal number: \(St=\frac{f}{U/D}={fcn}(Re)\).

The sole difference between these runs is the blockMeshFactor parameter, which increases block density and decreases timestep.

Probes were placed at \((x,y)=(5.5,\pm0.5)\), and a Fourier Transform of the nondimensionalized y-velocity was taken with respect to nondimensionalized time, s.t. the highest amplitude resultant frequency is the Strouhal number. Agreement between Strouhal numbers calculated for each probe was verified.

A table of results follows:

• Mesh | St | timestep
• 110_3 0.180995 0.000666667;
  
• 110_2 0.182648 0.000555556;
  
• 110_4 0.181818 0.000444444;

So clearly the approximations converge to a Strouhal number around 0.182.

Plots for a probe from each mesh follow, along with the Fourier Transform.

• Mesh 110_3

• Mesh 110_2

• Mesh 110_4

Looking at separation on Mesh 110_2, we see that it occurs at about 117 degrees, as predicted by Park et. al. (Numerical Solutions of Flow Past a Circular Cylinder at Reynolds Numbers up to 160, 1203).

The referenced plot from Park et. al.:

Steady Flow:

Drag Coefficient for Re=20:

Calculation of force on the cylinder can be done by implementing a simple control volume approach, by measuring the far-field pressure and velocity at the inlet and outlet as follows. Note all units are non-dimensionalized, and the force is per unit length (of the cylinder).

By momentum conservation, we have the following, where y is the vertical dimension perpendicular to the cylinder axis and the fluid’s general velocity. Note the inlet-outlet notation denotes a pair of integrals for the inlet and outlet, subtracted from each other.

\[F=\int_{inlet-outlet} p \,dy + \int_{inlet-outlet} \frac{\partial{\vec{p}}}{\partial{t}} \]

\[\implies F=\int_{inlet-outlet} p \,dy + \int_{inlet-outlet} \rho\|{\vec{u}}\|\vec{u}_x \partial{y}\]

Note this is trivial to calculate, since all the values are non-dimensionalized by OpenFoam.

The resulting force is 1.358 in units of \(\rho U^{2}L\), which becomes a Cd of 2.717.

Comparing this to results from Tritton (Physical Fluid Dynamics, 1977), and Heddleson et. al. (Summary of Drag Coefficients of Various Shaped Cylinders), we expect a Cd around 2.4 for a Reynolds number of 20, and our result becomes reasonable. Reference images from Tritton and Heddleson are attached below.